COPYRIGHTED MATERIAL. Introduction. 1.1 Signals, Noise and Information

Transcription

1 1 Introduction Signal processing is concerned with the efficient and accurate modelling, extraction, communication and utilisation of information, patterns and structures in a signal process Signal processing provides the theory, the methods and the tools for such purposes as the analysis and modelling of signals, extraction of information from signals, classification and recognition of patterns, synthesis and morphing of signals morphing is the creation of a new voice or image out of existing samples The applications of signal processing methods are very wide and include hi-fi audio, TV and radio, cellular mobile phones, voice recognition, vision, antenna-arrays, radar, sonar, geophysical exploration, medical electronics, bio-medical signal processing, physics and generally any system that is concerned with the communication or processing and retrieval of information Signal processing plays a central role in the development of new generations of mobile telecommunication and intelligent automation systems and in the efficient transmission, reception, decoding, organisation and retrieval of information content in search engines This chapter begins with a definition of signals, and a brief introduction to various signal processing methodologies We consider several key applications of digital signal processing in biomedical signal processing, adaptive noise reduction, channel equalisation, pattern classification/recognition, audio signal coding, signal detection, spatial processing for directional reception of signals, Dolby noise reduction, radar and watermarking This chapter concludes with an overview of the most basic process in a digital signal processing system, namely sampling and quantisation 11 Signals, Noise and Information Asignal is the variation of a quantity such as air pressure waves of sounds, colours of an image, depths of a surface, temperature of a body, current/voltage in a conductor or biological system, light, electromagnetic radio waves, commodity prices or volume and mass of an object A signal conveys information regarding one or more attributes of the source such as the state, the characteristics, the composition, the trajectory, the evolution or the intention of the source Hence, a signal is a means of conveying information regarding the past, the current or the future states of a variable For example, astrophysicists analyse the spectrum of signals, the light and other electromagnetic waves, emitted from distant stars or galaxies in order to deduce information about their movements, origins and COPYRIGHTED MATERIAL Advanced Digital Signal Processing and Noise Reduction c 28 John Wiley & Sons, Ltd Fourth Edition Saeed V Vaseghi

2 2 Introduction evolution Imaging radars calculate the round trip delay of reflected light or radio waves bouncing from the surface of the earth in order to produce maps of the earth A signal is rarely observed in isolation from a combination of noise, and distortion In fact noise and distortion are the fundamental sources of the limitations of: (a) the capacity, or equivalently the maximum speed, to send/receive information in a communication system, (b) the accuracy of measurements in signal processing and control systems and (c) the accuracy of decisions in pattern recognition As explained in Chapter 2 noise itself is a signal be it an unwanted signal that gives information on the state of the source of noise; for example the noise from a mechanical system conveys information on its working order A signal may be a function of one dimension, that is a function of one variable, such as speech or music whose amplitude fluctuations are a function of the time variable, or a signal can be multidimensional such as an image (ie reflected light intensity) which is a function of two-dimensional space or a video sequence which is a function of two-dimensional space and time Note that a photograph effectively projects a view of objects in three-dimensional space onto a two-dimensional image plane where depth information can be deduced from the shadows and gradients of colours The information conveyed in a signal may be used by humans or machines (eg computers or robots) for communication, forecasting, decision-making, control, geophysical exploration, medical diagnosis, forensics, etc The types of signals that signal processing systems deal with include text, image, audio, video, ultrasonic, subsonic, electromagnetic waves, medical, biological, thermal, financial or seismic signals Figure 11 illustrates a simplified overview of a communication system composed of an information source I(t) followed by: a system T[ ] for transformation of the information into variation of a signal x(t) that carries the information, a communication channel h[ ] for modelling the propagation of the signal from the transmitter to the receiver, additive channel and background noise n(t) that exists in every real-life system and a signal processing unit at the receiver for extraction of the information from the received signal Figure 11 Illustration of a communication and signal processing system In general, there is a mapping operation (eg modulation) that maps the output I(t) of an information source to the physical variations of a signal x(t) that carries the information over the channel, this mapping operator may be denoted as T[ ] and expressed as x(t) = T[I(t)] (11) The information source I(t) is normally discrete-valued whereas the signal x(t) that carries the information to a receiver may be continuous or discrete For example, in multimedia communication the information from a computer, or any other digital communication device, is in the form of a sequence of binary numbers (ones and zeros) which would need to be transformed into a physical quantity such as

3 Signal Processing Methods 3 voltage or current and modulated to the appropriate form for transmission in a communication channel such as a radio channel, telephone line or cable As a further example, in human speech communication the voice-generating mechanism provides a means for the speaker to map each discrete word into a distinct pattern of modulation of the acoustic vibrations of air that can propagate to the listener To communicate a word w, the speaker generates an acoustic signal realisation of the word x(t); this acoustic signal may be contaminated by ambient noise and/or distorted by a communication channel or room reverberations, or impaired by the speaking abnormalities of the talker, and received as the noisy, distorted and/or incomplete signal y(t) modelled as y(t) = h[x(t)]+n(t) (12) Where the function h[] models the channel distortion In addition to conveying the spoken word, the acoustic speech signal conveys information on the prosody (ie pitch intonation and stress patterns) of speech and the speaking characteristic, accent and the emotional state of the talker The listener extracts this information by processing the signal y(t) In past few decades, the theory and applications of digital signal processing have evolved to play a central role in the development of modern telecommunication and information technology systems Signal processing methods are central to efficient mobile communication, and to the development of intelligent man/machine interfaces in such areas as speech and visual pattern recognition for multimedia systems In general, digital signal processing is concerned with two broad areas of information theory: (1) Efficient and reliable coding, transmission, reception, storage and representation of signals in communication systems such as mobile phones, radio and TV (2) The extraction of information from noisy and/or incomplete signals for pattern recognition, detection, forecasting, decision-making, signal enhancement, control, automation and search engines In the next section we consider four broad approaches to signal processing 12 Signal Processing Methods Signal processing methods provide a variety of tools for modelling, analysis, coding, synthesis and recognition of signals Signal processing methods have evolved in algorithmic complexity aiming for the optimal utilisation of the available information in order to achieve the best performance In general, the computational requirement of signal processing methods increases, often exponentially, with the algorithmic complexity However, the implementation costs of advanced signal processing methods have been offset and made affordable by the consistent trend in recent years of a continuing increase in performance, coupled with a simultaneous decrease in the cost of signal processing hardware Depending on the method used, digital signal processing algorithms can be categorised into one or a combination of four broad categories These are transform-based signal processing, model-based signal processing, Bayesian statistical signal processing and neural networks are illustrated in Figure 12 These methods are described briefly in the following 121 Transform-Based Signal Processing The purpose of a transform is to express a signal or a system in terms of a combination of a set of elementary simple signals (such as sinusoidal signals, eigen vectors or wavelets) that lend themselves to relatively easy analysis, interpretation and manipulation Transform-based signal processing methods include Fourier transform, Laplace transform, z-transform and wavelet transforms

4 DSP Applications Communication Signal Processing Transmission/Reception/Storage Information Extraction, Content Processing Information Management, System Control Source Coding and Channel Coding Space-Time Array Processing Model Estimation Pattern Recognition Speech/music coding image/video coding, data compression, communication over noisy channels, channel equalisation, watermarking Antenna arrays, sonar, radar, microphone arrays mobile communication, biosignal enhancement Spectral analysis, radar and sonar signal processing, signal enhancement, geophysics exploration Speech recognition, music recognition image and character recognition, bio-signal information processing, search engines Figure 12 A broad categorisation of some of the most commonly used signal processing methods ICA = Independent Component Analysis, HOS = Higher order statistics Note that there may be overlap between different methods and also various methods can be combined

5 Signal Processing Methods 5 The most widely applied signal transform is the Fourier transform which is effectively a form of vibration analysis; a signal is expressed in terms of a combination of the sinusoidal vibrations that make up the signal Fourier transform is employed in a wide range of applications including popular music coders, noise reduction and feature extraction for pattern recognition The Laplace transform, and its discrete-time version the z-transform, are generalisations of the Fourier transform and describe a signal or a system in terms of a set of transient sinusoids with exponential amplitude envelopes In Fourier, Laplace and z-transform, the different sinusoidal basis functions of each transform all have the same duration and differ in terms of their frequency of vibrations In contrast wavelets are multi-resolution transforms in which a signal is described in terms of a combination of elementary waves of different dilations The set of basis functions in a wavelet is composed of contractions and dilations of a single elementary wave This allows non-stationary events of various durations in a signal to be identified and analysed Wavelet analysis is effectively a tree-structured filter bank analysis in which a set of high pass and low filters are used repeatedly in a binary-tree structure to split the signal progressively into a set of non-uniform sub-bands with different bandwidths 122 Source-Filter Model-Based Signal Processing Model-based signal processing methods utilise a parametric model of the signal generation process The parametric model normally describes the predictable structures and the expected patterns in the signal process, and can be used to forecast the future values of a signal from its past trajectory Model-based methods normally outperform non-parametric methods, since they utilise more information in the form of a model of the signal process However, they can be sensitive to the deviations of a signal from the class of signals characterised by the model The most widely used parametric model is the linear prediction model, described in Chapter 8 Linear prediction models have facilitated the development of advanced signal processing methods for a wide range of applications such as low-bit-rate speech coding in cellular mobile telephony, digital video coding, high-resolution spectral analysis, radar signal processing and speech recognition 123 Bayesian Statistical Model-Based Signal Processing Statistical signal processing deals with random processes; this includes all information-bearing signals and noise The fluctuations of a random signal, or the distribution of a class of random signals in the signal space, cannot be entirely modelled by a predictive equation, but it can be described in terms of the statistical average values, and modelled by a probability distribution function in a multidimensional signal space For example, as described in Chapter 8, a linear prediction model driven by a random signal can provide a source-filter model of the acoustic realisation of a spoken word However, the random input signal of the linear prediction model, or the variations in the characteristics of different acoustic realisations of the same word across the speaking population, can only be described in statistical terms and in terms of probability functions Bayesian inference theory provides a generalised framework for statistical processing of random signals, and for formulating and solving estimation and decision-making problems Bayesian methods are used for pattern recognition and signal estimation problems in applications such as speech processing, communication, data management and artificial intelligence In recognising a pattern or estimating a signal, from noisy and/or incomplete observations, Bayesian methods combine the evidence contained in the incomplete signal observation with the prior information regarding the distributions of the signals and/or the distributions of the parameters associated with the signals Chapter 4 describes Bayesian inference methodology and the estimation of random processes observed in noise

6 6 Introduction 124 Neural Networks Neural networks are combinations of relatively simple non-linear adaptive processing units, arranged to have a structural resemblance to the transmission and processing of signals in biological neurons In a neural network several layers of parallel processing elements are interconnected with a hierarchically structured connection network The connection weights are trained to memorise patterns and perform a signal processing function such as prediction or classification Neural networks are particularly useful in non-linear partitioning of a signal space, in feature extraction and pattern recognition, and in decision-making systems In some hybrid pattern recognition systems neural networks are used to complement Bayesian inference methods Since the main objective of this book is to provide a coherent presentation of the theory and applications of statistical signal processing, neural networks are not discussed here 13 Applications of Digital Signal Processing In recent years, the development and commercial availability of increasingly powerful and affordable digital computers has been accompanied by the development of advanced digital signal processing algorithms for a wide variety of applications such as noise reduction, telecommunication, radar, sonar, video and audio signal processing, pattern recognition, geophysics explorations, data forecasting, and the processing of large databases for the identification, extraction and organisation of unknown underlying structures and patterns Figure 13 shows a broad categorisation of some DSP applications This section provides a review of several key applications of digital signal processing methods In the following an overview of some applications of DSP is provided Note that these applications are by no means exhaustive but they represent a useful introduction 131 Digital Watermarking Digital watermarking is the embedding of a signature signal, ie the digital watermark, underneath a host image, video or audio signal Although watermarking may be visible or invisible, the main challenge in digital watermarking is to make the watermark secret and imperceptible (meaning invisible or inaudible) Watermarking takes its name from the watermarking of paper or money for security and authentication purposes Watermarking is used in digital media for the following purposes: (1) Authentication of digital image and audio signals The watermark may also include owner information, a serial number and other useful information (2) Protection of copyright/ownership of image and audio signals from unauthorised copying, use or trade (3) Embedding of audio or text signals into image/video signals for subsequent retrieval (4) Embedding a secret message into an image or audio signal Watermarking has to be robust to intended or unintended degradations and resistant to attempts at rendering it ineffective In particular watermarking needs to survive the following processes: (1) Changes in the sampling rate, resolution and format of the signal (2) Changes in the orientation of images or phase of the signals (3) Noise and channel distortion (4) Non-linear imperceptible changes of time/space scales For example non-linear time-warping of audio or non-linear warping of the dimensions of an image (5) Segmentation and cropping of the signals

7 DSP Applications Communication Signal Processing Transmission/Reception/Storage Information Extraction, Content Processing Information Management, System Control Source Coding and Channel Coding Space-Time Array Processing Model Estimation Pattern Recognition Speech/music coding image/video coding, data compression, communication over noisy channels, channel equalisation, watermarking Antenna arrays, sonar, radar, microphone arrays mobile communication, biosignal enhancement Spectral analysis, radar and sonar signal processing, signal enhancement, geophysics exploration Speech recognition, music recognition image and character recognition, bio-signal information processing, search engines Figure 13 A classification of the applications of digital signal processing

8 8 Introduction The simplest forms of watermarking methods, illustrated in Figure 14, exploit the time frequency structure of the signal together with the audio-visual perceptual characteristics of humans The watermark signal is hidden in the parts of the host signal spectrum, where it is invisible in the case of image signals or inaudible in the case of audio signals Discrete cosine transform or wavelet transform are commonly used for transforming the host signal to frequency-time domains As shown in Figure 14 the watermark is randomised and hidden using a secret key before it is embedded in the host signal This introduces and additional level of security Figure 14 A simplified illustration of frequency domain watermark embedding (top) and watermark retrieval (bottom) The secret key introduces an additional level of security Reproduced by permission of 28 Saeed V Vaseghi An example of invisible watermarking is shown in Figure 15The figure shows a host image and another image acting as the watermark together with the watermarked image and the retrieved watermark 132 Bio-medical, MIMO, Signal Processing Bio-medical signal processing is concerned with the analysis, denoising, synthesis and classification of bio-signals such as magnetic resonance images (MRI) of the brain or electrocardiograph (ECG) signals of the heart or electroencephalogram (EEG) signals of brain neurons An electrocardiograph signal is produced by recording the electrical voltage signals of the heart It is the main tool in cardiac electrophysiology, and has a prime function in the screening and diagnosis of cardiovascular diseases Electroencephalography is the neurophysiologic measurement of the electrical activity of the neurons in the brain picked up by electrodes placed on the scalp or, in special cases, on the cortex The resulting

9 Applications of Digital Signal Processing 9 Figure 15 Illustration of invisible watermarking of an image, clockwise from top-left: a picture of my son, the watermark, watermarked image and retrieved watermark The watermark may be damaged due to modifications such as a change of image coding format Reproduced by permission of 28 Saeed V Vaseghi signals are known as an electroencephalograph and represent a mix of electrical signals and noise from a large number of neurons The observations of ECG or EEG signals are often a noisy mixture of electrical signals generated from the activities of several different sources from different parts of the body The main issues in the processing of bio-signals, such as EEG or ECG, are the denoising, separation and identification of the signals from different sources An important bio-signal analysis tool, considered in Chapter 18, is known as independent component analysis (ICA) ICA is primarily used for separation of mixed signals in multi-source multi-sensor applications such as in ECG and EEG ICA is also used for beam forming in multiple-input multiple-output (MIMO) telecommunication The ICA problem is formulated as follows The observed signal vector x is assumed to be a linear mixture of M independent source signals s In a linear matrix form the mixing operation is expressed as x = As (13)

10 1 Introduction The matrix A is known as the mixing matrix or the observation matrix In many practical cases of interest all we have is the sequence of observation vectors [x(), x(1),, x(n 1)] The mixing matrix A is unknown and we wish to estimate a demixing matrix W to obtain an estimate of the original signal s This problem is known as blind source separation (BSS); the term blind refers to the fact that we have no other information than the observation x and an assumption that the source signals are independent of each other The demixing problem is the estimation of a matrix W such that ŝ =Wx (14) The details of the derivation of the demixing matrix are discussed in Chapter 18 on ICA Figure 16 shows an example of ECG signal mixture of the hearts of a pregnant mother and foetus plus other noise and interference Note that application of ICA results in separation of the mother and foetus heartbeats Also note that the foetus heartbeat rate is about 25 % faster than the mother s heart-beat rate 5 Sensor signals 5 Mother component 2 Foetal component Figure 16 Application of ICA to separation of mother and foetus ECG Note that signals from eight sensors are used in this example 133 Echo Cancellation Echo is the repetition of a signal back to the transmitter; either due to a coupling between the loudspeaker and microphone or due to a reflection of the transmitted signal from the points or surfaces where the characteristics of the medium through which the signal propagates changes significantly so as to impede the propagation of the signal in the original direction such that some of the signal energy is reflected back to the source

11 Applications of Digital Signal Processing 11 Modern telecommunication systems, Figure 17, connect a variety of voice-enabled terminals, such as fixed telephones, mobile phones, laptops etc via a variety of networks and relays including public switched telephone network (PSTN), satellites, cellular networks, voice over internet protocol (VoIP), wifi, etc Echo can severely affect the quality and intelligibility of voice conversation in telephone, teleconference, VoIP or cabin communication systems Hybrid echo Echo cancellers Mobile switching centre Acoustic echo (a) PSTN, Satellites, Cellular Networks, IP Networks (VoIP), Wifi etc (b) Figure 17 (a) Illustration of sources of echo in a mobile-to-landline system, (b) a modern communication network connects a variety of voice-enabled devices through a host of different telephone and IP networks Echo cancellation is an important aspect of the design of telecommunication systems such as conventional wire-line telephones, hands-free phones, cellular mobile (wireless) phones, teleconference systems, voice over internet (VoIP) and in-vehicle cabin communication systems There are two types of echo in a voice communication system (Figure 17(a)): (1) Acoustic echo due to acoustic coupling between the speaker and the microphone in hands-free phones, mobile phones and teleconference systems (2) Electrical line echo due to mismatch at the hybrid circuit connecting a two-wire subscriber line to a four-wire trunk line in the public switched telephone network Voice communication systems cannot function properly without echo cancellation systems A solution used in the early days was echo suppression However, modern communication systems employ adaptive echo cancellation systems that identify the echo path and synthesis a replica of the echo that is subtracted from the actual echo in order to remove the echo Echo cancellation is covered in Chapter 15

12 12 Introduction 134 Adaptive Noise Cancellation In speech communication from a noisy acoustic environment such as a moving car or train, or over a noisy telephone channel, the speech signal is observed in an additive random noise In signal measurement systems the information-bearing signal is often contaminated by noise from its surrounding environment The noisy observation y(m) can be modelled as y(m) = x(m) + n(m) (15) where x(m) and n(m) are the signal and the noise, and m is the discrete-time index In some situations, for example when using a mobile telephone in a moving car, or when using a radio communication device in an aircraft cockpit, it may be possible to measure and estimate the instantaneous amplitude of the ambient noise using a directional microphone The signal x(m) may then be recovered by subtraction of an estimate of the noise from the noisy signal Figure 18 shows a two-input adaptive noise cancellation system for enhancement of noisy speech In this system a directional microphone takes as input the noisy signal x(m) + n(m), and a second directional microphone, positioned some distance away, measures the noise α n(m + τ) The attenuation factor α and the time delay τ provide a rather over-simplified model of the effects of propagation of the noise to different positions in the space where the microphones are placed The noise from the second microphone is processed by an adaptive digital filter to make it equal to the noise contaminating the speech signal, and then subtracted from the noisy signal to cancel out the noise The adaptive noise canceller is more effective in cancelling out the low-frequency part of the noise, but generally suffers from the non-stationary character of the signals, and from the over-simplified assumption that a linear filter can model the diffusion and propagation of the noise sound in the space Noisy signal y(m) = x(m)+n(m) Noise (m+ ) z z z w w 1 w 2 w P-1 Adaptation algorithm Signal x(m) ^ Noise Estimation Filter Noise estimate n(m) ^ Figure 18 Configuration of a two-microphone adaptive noise canceller The adaptive filter delay elements (z 1 ) and weights w i model the delay and attenuation that signals undergo while propagating in a medium 135 Adaptive Noise Reduction In many applications, for example at the receiver of a telecommunication system, there is no access to the instantaneous value of the contaminating noise, and only the noisy signal is available In such cases the noise cannot be cancelled out, but it may be reduced, in an average sense, using the statistics of the signal and the noise process Figure 19 shows a bank of Wiener filters for reducing additive noise when only the noisy signal is available The filter bank coefficients attenuate each noisy signal frequency in inverse

13 Applications of Digital Signal Processing 13 Noisy signal y(m)=x(m)+n(m) y() y(1) y(2) y(n-1) Discrete Fourier Transform Y() Y(1) Y(2) Y(N-1) W W 1 W 2 W N -1 X() ^ X(1) ^ X(2) ^ X(N-1) ^ Inverse Discrete Fourier Transform Restored signal x() ^ x(1) ^ x(2) ^ x(n-1) ^ Signal and noise power spectra Wiener filter estimator Figure 19 A frequency domain Wiener filter for reducing additive noise proportion to the signal-to-noise ratio at that frequency The Wiener filter bank coefficients, derived in Chapter 6, are calculated from estimates of the power spectra of the signal and the noise processes 136 Blind Channel Equalisation Channel equalisation is the recovery of a signal distorted in transmission through a communication channel with a non-flat magnitude and/or a non-linear phase response When the channel response is unknown the process of signal recovery is called blind equalisation Blind equalisation has a wide range of applications, for example in digital telecommunications for removal of inter-symbol interference due to non-ideal channel and multi-path propagation, in speech recognition for removal of the effects of the microphones and the communication channels, in correction of distorted images, analysis of seismic data, de-reverberation of acoustic gramophone recordings etc In practice, blind equalisation is feasible only if some useful statistics of the channel input are available The success of a blind equalisation method depends on how much is known about the characteristics of the input signal and how useful this knowledge can be in the channel identification and equalisation process Figure 11 illustrates the configuration of a decision-directed equaliser This blind channel equaliser is composed of two distinct sections: an adaptive equaliser that removes a large part of the channel distortion, followed by a non-linear decision device for an improved estimate of the channel input The output of the decision device is the final estimate of the channel input, and it is used as the desired signal to direct the equaliser adaptation process Blind equalisation is covered in detail in Chapter Signal Classification and Pattern Recognition Signal classification is used in detection, pattern recognition and decision-making systems For example, a simple binary-state classifier can act as the detector of the presence, or the absence, of a known waveform in noise In signal classification, the aim is to design a minimum-error system for labelling a signal with one of a number of likely classes of signal

14 14 Introduction Channel noise n(m) x(m) Channel distortion H(f) f + y(m) Equaliser H inv (f) Adaptation algorithm f Error signal Decision device ^ x(m) Blind decision-directed equaliser Figure 11 Configuration of a decision-directed blind channel equaliser To design a classifier, a set of models are trained for the classes of signals that are of interest in the application The simplest form that the models can assume is a bank, or codebook, of waveforms, each representing the prototype for one class of signals A more complete model for each class of signals takes the form of a probability distribution function In the classification phase, a signal is labelled with the nearest or the most likely class For example, in communication of a binary bit stream over a band-pass channel, the binary phase-shift keying (BPSK) scheme signals the bit 1 using the waveform A c sin ω c t and the bit using A c sin ω c t At the receiver, the decoder has the task of classifying and labelling the received noisy signal as a 1 or a Figure 111 illustrates a correlation receiver for a BPSK signalling scheme The receiver has two correlators, each programmed with one of the two symbols representing the binary states for the bit 1 and the bit The decoder correlates the unlabelled input signal with each of the two candidate symbols and selects the candidate that has a higher correlation with the input Received noisy symbol Correlator for symbol "1" Correlator for symbol "" Corel(1) Corel() Decision device "1" if Corel(1) Corel() " " if Corel(1) < Corel() "1" Figure 111 Block diagram illustration of the classifier in a binary phase-shift keying demodulation Figure 112 illustrates the use of a classifier in a limited-vocabulary, isolated-word speech recognition system Assume there are V words in the vocabulary For each word a model is trained, on many different examples of the spoken word, to capture the average characteristics and the statistical variations of the word The classifier has access to a bank of V + 1 models, one for each word in the vocabulary and an additional model for the silence periods In the speech recognition phase, the task is to decode and label an acoustic speech feature sequence, representing an unlabelled spoken word, as one of the V likely words or silence For each candidate word the classifier calculates a probability score and selects the word with the highest score

15 Applications of Digital Signal Processing 15 Word model f Y M (Y 1 ) 1 likelihood of 1 Word model 2 Speech signal Feature extractor Feature sequence Y f Y M (Y 2 ) Word model f Y M (Y V ) likelihood of 2 V likelihood of v Most likely word selector ML Silence model sil f Y M (Y sil ) likelihood of sil Figure 112 Configuration of speech recognition system, f Y M (Y M i ) is the likelihood of the model M i given an observation sequence Y 138 Linear Prediction Modelling of Speech Linear predictive models (introduced in Chapter 8) are widely used in speech processing applications such as low-bit-rate speech coding in cellular telephony, speech enhancement and speech recognition Speech is generated by inhaling air into the lungs, and then exhaling it through the vibrating glottis cords and the vocal tract The random, noise-like, air flow from the lungs is spectrally shaped and amplified by the vibrations of the glottal cords and the resonance of the vocal tract The effect of the vibrations of the glottal cords and the resonance of the vocal tract is to shape the frequency spectrum of speech and introduce a measure of correlation and predictability on the random variations of the air from the lungs Figure 113 illustrates a source-filter model for speech production The source models the lungs Pitch period Random source Excitation Glottal (pitch) model P(z) Vocal tract model H(z) Speech Figure 113 Linear predictive model of speech

16 16 Introduction and emits a random excitation signal which is filtered, first by a pitch filter model of the glottal cords and then by a model of the vocal tract The main source of correlation in speech is the vocal tract modelled by a linear predictor A linear predictor is an adaptive filter that forecasts the amplitude of the signal at time m, x(m), using a linear combination of P previous samples [x(m 1),, x(m P)] as ˆx(m) = P a k x(m k) (16) k=1 where ˆx(m) is the prediction of the signal x(m), and the vector a T =[a 1,, a P ] is the coefficients vector of a predictor of order P The prediction error e(m), ie the difference between the actual sample x(m) and its predicted value ˆx(m), is defined as e(m) = x(m) P a k x(m k) (17) In speech processing, the prediction error e(m) may also be interpreted as the random excitation or the so-called innovation content of x(m) From Equation (17) a signal generated by a linear predictor can be synthesised as P x(m) = a k x(m k) + e(m) (18) k=1 Linear prediction models can also be used in a wide range of applications to model the correlation or the movements of a signal such as the movements of scenes in successive frames of video k=1 139 Digital Coding of Audio Signals In digital audio, the memory required to record a signal, the bandwidth and power required for signal transmission and the signal-to-quantisation-noise ratio are all directly proportional to the number of bits per sample The objective in the design of a coder is to achieve high fidelity with as few bits per sample as possible, at an affordable implementation cost Audio signal coding schemes utilise the statistical structures of the signal, and a model of the signal generation, together with information on the psychoacoustics and the masking effects of hearing In general, there are two main categories of audio coders: model-based coders, used for low-bit-rate speech coding in applications such as cellular telephony; and transform-based coders used in high-quality coding of speech and digital hi-fi audio Figure 114 shows a simplified block diagram configuration of a speech coder decoder of the type used in digital cellular telephones The speech signal is modelled as the output of a filter excited by a random signal The random excitation models the air exhaled through the lungs, and the filter models the vibrations of the glottal cords and the vocal tract At the transmitter, speech is segmented into blocks of about 3 ms long during which speech parameters can be assumed to be stationary Each block of speech samples is analysed to extract and transmit a set of excitation and filter parameters that can be used to synthesise the speech At the receiver, the model parameters and the excitation are used to reconstruct the speech A transform-based coder is shown in Figure 115 The aim of transformation is to convert the signal into a form where it lends itself to a more convenient and useful interpretation and manipulation In Figure 115 the input signal may be transformed to the frequency domain using a discrete Fourier

17 Applications of Digital Signal Processing 17 Speech x(m) (a) Source coder Model-based speech analysis Pitch and vocal-tract coefficients Excitation e(m) Scalar quantiser Vector quantiser Synthesiser coefficients Excitation address Excitation address Pitch coefficients Vocal-tract coefficients Excitation codebook Pitch filter Vocal-tract filter Reconstructed speech (b) Source decoder Figure 114 Block diagram configuration of a model-based speech (a) coder and (b) decoder Input signal Binary coded signal Reconstructed signal ^ x() X() n bps X() x() ^ ^ x(1) X(1) n 1 bps X(1) x(1) ^ ^ X(2) n X(2) x(2) 2 bps x(2) ^ x(n-1) Transform T X(N-1) Encoder n N-1 bps Decoder ^ X(N-1) Inverse Transform T -1 x(n-1) ^ Figure 115 Illustration of a transform-based coder transform or a discrete cosine transform or a filter bank Three main advantages of coding a signal in the frequency domain are: (1) The frequency spectrum of a signal has a relatively well-defined structure, for example most of the signal power is usually concentrated in the lower regions of the spectrum (2) A relatively low-amplitude frequency would be masked in the near vicinity of a large-amplitude frequency and can therefore be coarsely encoded without any audible degradation (3) The frequency samples are orthogonal and can be coded independently with different precisions The number of bits assigned to each frequency of a signal is a variable that reflects the contribution of that frequency to the reproduction of a perceptually high-quality signal In an adaptive coder, the allocation of bits to different frequencies is made to vary with the time variations of the power spectrum of the signal 131 Detection of Signals in Noise In the detection of signals in noise, the aim is to determine if the observation consists of noise alone, or if it contains a signal The noisy observation y(m) can be modelled as y(m) = b(m)x(m) + n(m) (19)

18 18 Introduction where x(m) is the signal to be detected, n(m) is the noise and b(m) is a binary-valued state indicator sequence such that b(m) = 1 indicates the presence of the signal x(m) and b(m) = indicates that the signal is absent If the signal x(m) has a known shape, then a correlator or a matched filter can be used to detect the signal as shown in Figure 116 The impulse response h(m) of the matched filter for detection of a signal x(m) is the time-reversed version of x(m) given by h(m) = x(n 1 m) m N 1 (11) where N is the length of The output of the matched filter is given by N 1 z(m) = h(k)y(m k) (111) k= y(m)=x(m)+n(m) Matched filter h(m) = x(n 1 m) z(m) Threshold comparator ^ b(m) Figure 116 Configuration of a matched filter followed by a threshold comparator for detection of signals in noise The matched filter output is compared with a threshold and a binary decision is made as { 1 if abs ( z(m) ) threshold ˆb(m) = otherwise (112) where ˆb(m) is an estimate of the binary state indicator sequence b(m), and it may be erroneous in particular if the signal-to-noise ratio is low Table 11 lists four possible outcomes that together b(m) and its estimate ˆb(m) can assume The choice of the threshold level affects the sensitivity of the detector The higher the threshold, the less the likelihood that noise would be classified as signal, so the false alarm rate falls, but the probability of misclassification of signal as noise increases The risk in choosing a threshold value θ can be expressed as R (Threshold = θ) = P FalseAlarm (θ) + P Miss (θ) (113) The choice of the threshold reflects a trade-off between the misclassification rate P Miss (θ) and the false alarm rate P FalseAlarm (θ) Table 11 Four possible outcomes in a signal detection problem ˆb(m) b(m) Detector decision Signal absent (Correct) 1 Signal absent (Missed) 1 Signal present (False alarm) 1 1 Signal present (Correct) 1311 Directional Reception of Waves: Beam-forming Beam-forming is the spatial processing of plane waves received by an array of sensors such that the waves incidents at a particular spatial angle are passed through, whereas those arriving from other directions are attenuated Beam-forming is used in radar and sonar signal processing (Figure 117) to steer the reception of signals towards a desired direction, and in speech processing for reducing the effects of ambient noise

19 Applications of Digital Signal Processing 19 Figure 117 Sonar: detection of objects using the intensity and time delay of reflected sound waves Array of Microphones z 1 Array of adaptive FIR filters z 1 W 1,1 W 1,P 1 d W 1, + Incident plane wave 1 z 1 W 2, W 2,1 + z 1 W 2,P 1 Output N-1 z 1 z 1 W N 1, W N 1,1 + W N 1,P 1 Figure 118 Illustration of a beam-former, for directional reception of signals To explain the process of beam-forming, consider a uniform linear array of sensors as illustrated in Figure 118 The term linear array implies that the array of sensors is spatially arranged in a straight line and with equal spacing d between the sensors Consider a sinusoidal far-field plane wave with a frequency F propagating towards the sensors at an incidence angle of θ as illustrated in Figure 118

20 2 Introduction The array of sensors samples the incoming wave as it propagates in space The time delay for the wave to travel a distance of d between two adjacent sensors is given by τ = dcos(θ) (114) c where c is the speed of propagation of the wave in the medium The phase difference corresponding to a delay of τ is given by φ = 2π τ d cos θ = 2πF (115) T c where T is the period of the sine wave By inserting appropriate corrective time delays in the path of the samples at each sensor, and then averaging the outputs of the sensors, the signals arriving from the direction θ will be time-aligned and coherently combined, whereas those arriving from other directions will suffer cancellations and attenuations Figure 118 illustrates a beam-former as an array of digital filters arranged in space The filter array acts as a two-dimensional space-time signal processing system The space filtering allows the beam-former to be steered towards a desired direction, for example towards the direction along which the incoming signal has the maximum intensity The phase of each filter controls the time delay, and can be adjusted to coherently combine the signals The magnitude frequency response of each filter can be used to remove the out-of-band noise 1312 Space-Time Signal Processing Conventionally transmission resources are shared among subscribers of communication systems through the division of time and frequency leading to such resource-sharing schemes as time division multiple access or frequency division multiple access Space provides a valuable additional resource that can be used to improve both the communication capacity and quality for wireless communication systems Space-time signal processing refers to signal processing methods that utilise simultaneous transmission and reception of signals through multiple spatial routes The signals may arrive at the destinations at different times or may use different time slots Space-time signal processing, and in particular the division of space among different users, is an important area of research and development for improving the system capacity in the new generations of high-speed broadband multimedia mobile communication systems For example, in mobile communication the multi-path effect, where a radio signal propagates from the transmitter to the receiver via a number of different paths, can be used to advantage in space-time signal processing The multiple noisy versions of a signal, arriving via different routes with different noise and distortions, are processed and combined such that the signal components add up constructively and become stronger compared to the random uncorrelated noise The uncorrelated fading that the signals suffer in their propagation through different routes can also be mitigated The use of transmitter/receiver antenna arrays for beam-forming allows the division of the space into narrow sectors such that the same frequencies, in different narrow spatial sectors, can be used for simultaneous communication by different subscribers and/or different spatial sectors can be used to transmit the same information in order to achieve robustness to fading and interference In fact combination of space and time can provide a myriad of possibilities, as discussed in Chapter 19 on mobile communication signal processing Note that the ICA method, described in Section 132 and Chapter 18, is often used in space-time signal processing for separation of multiple signals at the receiver 1313 Dolby Noise Reduction Dolby noise reduction systems work by boosting the energy and the signal-to-noise ratio of the highfrequency spectrum of audio signals The energy of audio signals is mostly concentrated in the lowfrequency part of the spectrum (below 2 khz) The higher frequencies that convey quality and sensation

21 Applications of Digital Signal Processing 21 have relatively low energy, and can be degraded even by a low amount of noise For example when a signal is recorded on a magnetic tape, the tape hiss noise affects the quality of the recorded signal On playback, the higher-frequency parts of an audio signal recorded on a tape have smaller signal-to-noise ratio than the low-frequency parts Therefore noise at high frequencies is more audible and less masked by the signal energy Dolby noise reduction systems broadly work on the principle of emphasising and boosting the low energy of the high-frequency signal components prior to recording the signal When a signal is recorded it is processed and encoded using a combination of a pre-emphasis filter and dynamic range compression At playback, the signal is recovered using a decoder based on a combination of a de-emphasis filter and a decompression circuit The encoder and decoder must be well matched and cancel each other out in order to avoid processing distortion Dolby developed a number of noise reduction systems designated Dolby A, Dolby B and Dolby C These differ mainly in the number of bands and the pre-emphasis strategy that that they employ Dolby A, developed for professional use, divides the signal spectrum into four frequency bands: band 1 is low-pass and covers Hz to 8 Hz; band 2 is band-pass and covers 8 Hz to 3 khz; band 3 is high-pass and covers above 3 khz; and band 4 is also high-pass and covers above 9 khz At the encoder the gain of each band is adaptively adjusted to boost low-energy signal components Dolby A provides a maximum gain of 1 to 15 db in each band if the signal level falls 45 db below the maximum recording level The Dolby B and Dolby C systems are designed for consumer audio systems, and use two bands instead of the four bands used in Dolby A Dolby B provides a boost of up to 1 db when the signal level is low (less than 45 db than the maximum reference) and Dolby C provides a boost of up to 2 db as illustrated in Figure Relative gain (db) Frequency (khz) Figure 119 Illustration of the pre-emphasis response of Dolby C: up to 2 db boost is provided when the signal falls 45 db below maximum recording level 1314 Radar Signal Processing: Doppler Frequency Shift Figure 12 shows a simple diagram of a radar system that can be used to estimate the range and speed of an object such as a moving car or a flying aeroplane A radar system consists of a transceiver (transmitter/receiver) that generates and transmits sinusoidal pulses at microwave frequencies The signal travels with the speed of light and is reflected back from any object in its path The analysis of the received echo provides such information as range, speed and acceleration The received signal has the form x(t) = A(t) cos{ω [t 2r(t)/c]} (116)

22 22 Introduction where A(t), the time-varying amplitude of the reflected wave, depends on the position and the characteristics of the target, r(t) is the time-varying distance of the object from the radar and c is the velocity of light The time-varying distance of the object can be expanded in a Taylor series as r(t) = r +ṙt + 1 2! rt ! r t 3 + (117) where r is the distance, ṙ is the velocity, r is the acceleration etc Approximating r(t) with the first two terms of the Taylor series expansion we have Substituting Equation (118) in Equation (116) yields r(t) r +ṙt (118) x(t) = A(t) cos[(ω 2ṙω /c)t 2ω r /c] (119) Antenna Control Tranceiver cos( o t) cos( o (t-2r(t)/c)) DSP system r=5t c Display and computer Figure 12 Illustration of a radar system Note that the frequency of reflected wave is shifted by an amount ω d = 2ṙω /c (12) This shift in frequency is known as the Doppler frequency If the object is moving towards the radar then the distance r(t) is decreasing with time, ṙ is negative, and an increase in the frequency is observed Conversely if the object is moving away from the radar then the distance r(t) is increasing, ṙ is positive, and a decrease in the frequency is observed Thus the frequency analysis of the reflected signal can reveal information on the direction and speed of the object The distance r is given by r = 5 T c (121) where T is the round-trip time for the signal to hit the object and arrive back at the radar and c is the velocity of light 14 A Review of Sampling and Quantisation Digital signal processing involves the processing of signals by a computer or by a purpose-built signal processing microchip The signal is stored in the computer s memory in a binary format in terms of a

23 A Review of Sampling and Quantisation 23 sequence of n-bit words Hence, to digitally process signals that are not already in a digital format, the signals need to be converted into a digital format that can be stored and processed in a computing device Sampling and quantisation are the first two steps in all digital signal processing and digital communication systems which have analogue inputs Most signals such as speech, image and electromagnetic waves, are not naturally in a digital format but need to be digitised (ie sampled and quantised) for subsequent processing and storage in a digital system such as in a computer or in a mobile DSP chip or a digital music player A signal needs to be sampled at a rate of more than twice the highest frequency content of the signal; otherwise the sampling process will result in loss of information and distortion Hence, prior to sampling, the input signal needs to be filtered by an anti-aliasing filter to remove the unwanted signal frequencies above a preset value of less than half the sampling frequency Each sample value is subsequently quantised to the nearest of 2 n quantisation levels and coded with an n-bit word The digitisation process should be performed such that the original continuous signal can be recovered from its digital version with no loss of information, and with as high a fidelity as is required in an application A digital signal is a sequence of discrete real-valued or complex-valued numbers, representing the fluctuations of an information-bearing quantity with time, space or some other variable The most elementary unit of a discrete-time (or discrete-space) signal is the unit-sample signal δ(m) defined as { 1 m = δ(m) = (122) m = where m is the discrete-time index A digital signal x(m) can be expressed as the sum of a number of amplitude-scaled and time-shifted unit samples as x(m) = x(k)δ(m k) (123) k= Figure 121 illustrates a discrete-time signal and its continuous-time envelope Many signals such as speech, music, image, video, radar, sonar and bio-signals and medical signals are analogue, in that in their original form they appear to vary continuously with time (and/or space) and are sensed by analogue sensors such as microphones, optical devices and antennas Other signals such as stock market prices are inherently discrete-time and/or discrete amplitude signals Continuous signals are termed analogue signals because their fluctuations with time are analogous to the variations of the signal source For digital processing of continuous signals, the signals are first sampled and then each sample is converted into an n-bit binary digit The sampling and digitisation process should be performed such that the original continuous signal can be recovered from its digital version with no loss of information, and with as high a fidelity as is required in an application Discrete time m Figure 121 A discrete-time signal and its continuous-time envelope of variation

24 24 Introduction Analogue-to-digital conversion, that is the conversion of an analogue signal into a sequence of n-bit words, consists of the two basic steps of sampling and quantisation: (1) Sampling The first step is to sample a signal to produce a discrete-time and/or discrete-space signal The sampling process, when performed with sufficiently high frequency (greater than twice the highest frequency), can capture the fastest fluctuations of the signal, and can be a loss-less operation in that the original analogue signal can be recovered through interpolation of the sampled sequence (2) Quantisation The second step is quantisation of each sample value into an n-bit word Quantisation involves some irrevocable errors and possible loss of information However, in practice the quantisation error (aka quantisation noise) can be made negligible by using an appropriately high number of bits as in a digital audio hi-fi Figure 122 illustrates a block diagram configuration of a digital signal processor with an analogue input The anti-aliasing low-pass filter (LPF) removes the out-of-band signal frequencies above a pre-selected cut-off frequency which should be set to less than half the intended sampling frequency The sample-andhold (S/H) unit periodically samples the signal to convert the continuous-time signal into a discrete-time, continuous-amplitude signal Analog input y(t) LPF & S/H y a (m) y(m) x(m) x x(t) Digital signal a (m) ADC DAC LPF processor Figure 122 Configuration of a digital signal processing system with analogue input and output The analogue-to-digital converter (ADC) follows the S/H unit and maps each continuous-amplitude sample into an n-bit word After the signal is processed, the digital output of the processor can be converted back into an analogue signal using a digital-to-analogue converter (DAC) and a low-pass filter as illustrated in Figure 122 Figure 123 shows a sample-and-hold circuit diagram where a transistor switch is turned on and off thereby allowing the capacitor to charge up or down to the level of the input signal during the on periods and then holding the sample value during the off period R 2 x(t) R1 x(mt s ) T s C Figure 123 A simplified sample-and-hold circuit diagram; when the switch closes the capacitor charges or discharges to the input level 141 Advantages of Digital Format The advantages of the digital format are as follows: (1) Digital devices such as mobile phones are pervasive (2) Transmission bandwidth and storage space savings Digital compression techniques, such as MP3, can be used to compress a digital signal When combined with error-control coding and efficient

25 A Review of Sampling and Quantisation 25 digital modulation methods the required overall bandwidth is less than that of say an FM-modulated analogue signal of similar noise robustness and quality There is a similar reduction in storage requirement (3) Power savings Power saving depends on the compression rate and the modulation method In general digital systems can achieve power efficiency compared with analogue systems (4) Noise robustness Digital waveforms are inherently robust to noise and additional robustness can be provided through error-control coding methods (5) Security Digital systems can be encrypted for security, and in particular the code division multiple access (CDMA) method, employed for sharing of time/bandwidth resources in mobile phone networks, is inherently secure (6) Recovery and restoration Digital signals are more amenable to recovery of lost segments (7) Noise reduction Digital noise reduction methods can be used to substantially reduce noise and interference and hence improve the perceived quality and intelligibility of a signal (8) Editing and mixing of audio/video and other signals in digital format is relatively easy (9) Internet and multimedia systems Digital communication, pattern recognition, Internet and multimedia communication would not have been possible without the digital format 142 Digital Signals Stored and Transmitted in Analogue Format Digital signals are actually stored and transmitted in analogue format For example, a binary-state transistor stores a one or a zero as a quantity of electronic charge, in bipolar baseband signalling a 1 or a is signalled with a pulse of ±V volts and in digital radio-frequency signalling binary digits are converted to modulated sinusoidal carriers for transmission over the airwaves Also the digital data on a CD track consists of a sequence of bumps of micro to nanometre size arranged as a single, continuous, long spiral track of data 143 The Effect of Digitisation on Signal Bandwidth In its simplest form each binary bit ( 1 or ) in a bit-stream representation of a signal can be viewed as pulse of duration T seconds, resulting in a bit rate of r b = 1/T bps Using the Fourier transform, it can be shown that the bandwidth of such a pulse sequence is about 2/T = 2r b Hz For example, the digitisation of stereo music at a rate of 441 khz and with each sample quantised to 16 bits generates a bit rate r b of (2 channels 44 1 samples/second 16 bits per sample) kbps This would require a bandwidth of 2r b = khz However, using advanced compression and modulation methods the number of bits per second and the required bandwidth can be greatly reduced by a factor of more than Sampling a Continuous-Time Signal Figure 124 illustrates the process of sampling of a continuous-time signal in time and its effect on the frequency spectrum of the signal In time-domain a sampled signal can be modelled as the product of a continuous-time signal x(t) multiplied by a periodic impulse train sampler p(t) as x sampled (t) = x(t)p(t) = x(t) δ(t mt s ) m= (124)

26 26 Introduction where T s is the sampling interval (the sampling frequency is F s = 1/T s ), δ(t) is the discrete-time delta (unit-sample) function and the sampling train function p(t) is defined as p(t) = δ(t mt s ) (125) m= The spectrum, P(f ), of a periodic train of sampling impulses in time p(t), is a periodic train of impulses in frequency given by P(f ) = δ(f kf s ) (126) k= where F s = 1/T s is the sampling frequency Time domain x(t) Frequency domain X( f ) t B 2B B f Impulse-train-sampling function P ( f ) ( f kf s ) k Ts t F s F s =1/Ts = = x p (t) Impulse-train-sampled Xp( f ) signal f t F s /2 F s /2 f sh(t) Sample-and-hold function SH( f ) Ts = t F s = F s X sh (t) X( f ) S/H-sampled signal f t F s /2 F s /2 f Figure 124 A sample-and-hold signal is modelled as an impulse-train sampling followed by convolution with a rectangular pulse

27 A Review of Sampling and Quantisation 27 Since multiplication of two time-domain signals is equivalent to the convolution of their frequency spectra we have X sampled (f ) = FT[x(t)p(t)]=X(f ) P(f ) = X(f kf s ) (127) where the operator FT[] denotes the Fourier transform Note from Equation (127) that the convolution of a signal spectrum X(f ) with each impulse δ(f kf s ), shifts X(f ) and centres it on kf s Hence, Equation (127) shows that the sampling of a signal x(t) results in a periodic repetition of its spectrum X(f ) with the images of the baseband spectrum X(f ) centred on frequencies ±F s, ±2F s, as shown in Figure 124 Note in Figure 124 that a sample-and-hold process produces a sampled signal which is in the shape of an amplitude-modulated staircase function Also note that the sample-and-hold staircase function can itself be modelled as the output of a filter, with a rectangular impulse response, excited by an idealised sampling impulse train as shown in Figure 124 k= 145 Aliasing Distortion The process of sampling results in a periodic repetition of the spectrum of the original signal When the sampling frequency F s is higher than twice the maximum frequency content of the signal B Hz (ie F s > 2B), then the repetitions ( images ) of the signal spectra are separated as shown in Figure 124 In this case, the analogue signal can be recovered by passing the sampled signal through an analogue low-pass filter with a cut-off frequency of just above B Hz If the sampling frequency is less than 2B (ie F s < 2B), then the adjacent repetitions of the spectrum overlap and in this case the original spectrum cannot be recovered The distortion, due to an insufficiently high sampling rate, is irrevocable and is known as aliasing Note in Figure 125 that the aliasing distortion results in the high frequency components of the signal folding and appearing at the lower frequencies, hence the name aliasing Figure 126 shows the sum of two sine waves sampled at above and below the Nyquist sampling rate Note that below the Nyquist rate a frequency of F may appear at kf s + F where k is an integer, as shown in Figure 126 Base-band spectrum X(f) High frequency spectrum aliasing into low frequency parts Low frequency spectrum aliasing into high frequency parts -2Fs -Fs Fs 2Fs Frequency Figure 125 Aliasing distortion results from the overlap of spectral images (dashed curves) with the baseband spectrum Note high frequency aliases itself as low frequency and vice versa In this example the signal is sampled at half the required rate 146 Nyquist Sampling Theorem The above observation on aliasing distortion is the basis of the Nyquist sampling theorem, which states: a band-limited continuous-time signal, with highest frequency content (bandwidth) of B Hz, can be

28 28 Introduction Original signal 2 x 14 Original signal 1-1 Frequency Original signal down sampled but NOT filtered Time 5 Spectrogram of down sampled signal 1-1 Frequency Original signal down sampled but NOT filtered 1-1 Frequency Time Spectrogram of down sampled signal Time Figure 126 Illustration of aliasing Top panel: the sum of two sinewaves, the assumed frequencies of the sinewaves are 62 Hz and 12 4 Hz, the sampling frequency is 4 Hz Middle panel: the sine waves down-sampled by a factor of 4 to a sampling frequency 1 Hz; note the aliased frequencies appear at 1 62 = 38 Hz and = 24 Hz Bottom panel: the sine waves down-sampled by a factor of 8 to a sampling frequency of 5 Hz; note the aliased frequencies appear at = 12 Hz and at = 24 Hz recovered from its samples provided that the sampling frequency F s is greater than 2B samples per second so that there is no aliasing Note that the sampling frequency F s needs to be greater than 2B to avoid aliasing distortion and to allow frequency space for the transition band of a low-pass filter which is used to recover the original (baseband) continuous signal from its sampled version In practice sampling is achieved using an electronic switch that allows a capacitor to charge or discharge to the level of the input voltage once every T s seconds as illustrated in Figure 123 The sample-andhold signal can be modelled as the output of a filter with a rectangular impulse response, and with the impulse-train-sampled signal as the input as illustrated in Figure Quantisation Quantisation is the process of converting each continuous-valued sample of a signal into a discrete value sample that can be assigned a unique digital codeword For digital signal processing, discrete-time continuous-amplitude samples, from the sample-and-hold, are quantised and mapped into n-bit binary code words before being stored and processing Figure 127 illustrates an example of the quantisation of a signal into four discrete quantisation levels with each quantisation level represented by a 2-bit codeword For quantisation to n-bit codewords, the

29 A Review of Sampling and Quantisation 29 +V x(mt) Continuous amplitude samples Discrete amplitude samples 11 2V 1 1 V Figure 127 Illustration of offset-binary scalar quantisation amplitude range of the signal is divided into 2 n quantisation levels Each continuous-amplitude sample is quantised to the nearest quantisation level and then mapped to the n-bit binary code assigned to that level Quantisation is a many-to-one mapping; this means that all the infinite number of values that fall within the continuum of the infinite values of a quantisation band are mapped to one single value at the centre of the band The mapping is hence an irreversible process in that we cannot recover the exact value of the quantised sample The mapping between an analogue sample x a (m) and its quantised value x(m) can be expressed as x(m) = Q [x a (m)] (128) where Q[ ] is the quantising function The performance of a quantiser is measured by signal-to-quantisation noise ratio (SQNR) The quantisation noise is defined as the difference between the analogue value of a sample and its quantised value as e(m) = x(m) x a (m) (129) Now consider an n-bit quantiser with an amplitude range of ±V volts The quantisation step size is Δ = 2V/2 n Assuming that the quantisation noise is a zero-mean random process with a uniform probability distribution (ie a probability of 1/Δ and with an amplitude range of ±Δ/2) we can express the noise power as E[e 2 (m)]= /2 /2 p ( e(m) ) e 2 (m)de(m) = 1 = 2 12 = V 2 2 2n 3 P signal /2 /2 e 2 (m)de(m) (13) where E [ ] is the expectation or averaging operator and the function p(e(m)) = 1/Δ, shown in Figure (128), is the uniform probability density function of the noise and Δ = 2V2 n Using Equation (13) the SQNR is given by ( [ E x2 (m) ] ) ( ) Psignal SQNR(n) = 1 log 1 = 1 log E [e 2 (m)] 1 V 2 2 2n /3 ( ) V 2 = 1 log log log 1 2 2n (131) = 477 α + 6 n

30 3 Introduction where P signal is the mean signal power, and α is the ratio in decibels of the peak signal power V 2 to the mean signal power P signal, which for a sine wave α is 3 Therefore, from Equation (131) every additional bit in an analogue-to-digital converter results in a 6 db improvement in signal-to-quantisation noise ratio pem (( )) e( m) Figure 128 Illustration of the uniform probability distribution of the quantization noise 148 Non-Linear Quantisation, Companding A uniform quantiser is only optimal, in the sense of achieving the minimum mean squared error, when the input signal is uniformly distributed within the full range of the quantiser, so that the uniform probability distribution of the signal sample values and the uniform distribution of the quantiser levels are matched and hence different quantisation levels are used with equal probability When a signal has a non-uniform probability distribution then a non-uniform quantisation scheme matched to the probability distribution of the signal is more appropriate This can also be achieved through a transformation of the input signal to change the distribution of the input signal towards a uniform distribution prior to application of a uniform quantiser For speech signals, non-uniform qunatisation is achieved through a logarithmic compression of speech, a process known as companding, Figure 129 Companding (derived from compressing-expanding) refers to the process of first compressing an analogue signal at the transmitter, and then expanding this signal back to its original size at the receiver During the companding process, continuous-amplitude input samples are compressed logarithmically and then quantised and coded using a uniform quantiser The assumption is that speech has an exponential distribution and that the logarithm of speech has a more uniform distribution Compressed output Uniform input Figure 129 Illustration of the compression curves of A-law and u-law quantisers Note that the curves almost coincide and appear as one

31 A Review of Sampling and Quantisation 31 Figure 13 shows the effect of logarithmic compression on the distribution of a Gaussian signal Note from Figure 13(b) that the distribution of the Gaussian signal is more spread after logarithmic compression Figure 131 shows three sets of plots of speech and their respective histogram for speech quantised with 16 bits uniform quantisation, 8 bits uniform quantisation and logarithmic compression Number of occurence Signal amplitude (a) Number of occurence Signal amplitude (b) Figure 13 (a) The histogram of a Gaussian input signal to a u-law logarithmic function, (b) the histogram of the output of the u-law function 4 x 14 Original signal with 16 bit quantisation x 1 4 Signal converted to 8 bit quantisation x x 14 Logarithmic compression with 8 bit quantisation x x 14 Histogram of the original signal x x 14 Histogram of signal converted to 8 bit quantisation Histogram of logarithmic compression with 8 bit quantisation Figure 131 From top panel, plots of speech and their histograms quantised with: 16 bits uniform, 8 bits uniform and 8 bits logarithmic respectively